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What is Probability? 

Probability mainly means possibility and it is an important branch of mathematics that aims at dealing with occurrence of an event. The value in this case is mainly expressed from zero to one. Probability was introduced as a part of Mathematics for the purpose of predicting the likelihood of events that can happen. Meaning of probability is based on the extent up to which an event is most likely to happen. The basic theory of probability is also used as a part of probability distribution in which an individual can learn possibilities of the outcomes related to an experiment.

Terminology of Probability Theory

An experiment is considered to be a planned operation that is mainly carried out under the controlled conditions. If result of an experiment is not predetermined then it is considered to be a chance experiment. The result of the experiment is termed as an outcome. The sample space of the experiment is the set of the possible outcomes. The three different ways of representing a sample space include, listing the possible outcomes, creating a tree diagram, or creating a Venn diagram. For example, if an individual flips one fair coin, SS = {HH, TT} where HH = heads and TT = tails are the outcomes. Event can be defined as the combination of various outcomes. Equally likely mainly means that each of the outcomes of a particular experiment can occur with equal probability. 

Probability Formula

The probability formula is mainly used for computing the probability of an event that can occur. The process of recalling the likelihood of an event happening is termed probability.

The formula of the probability of an event is:

P(A) = Number of favourable outcomes/ Total number of favourable outcomes

P(A) = n(A)/n(S)

Different Probability Formulas

List of all the Probability Formulas in Maths
Probability Range	0 ≤ P(A) ≤ 1
Rule of Addition	P(A∪B) = P(A) + P(B) – P(A∩B)
Rule of Complementary Events	P(A’) + P(A) = 1
Disjoint Events	P(A∩B) = 0
Independent Events	P(A∩B) = P(A) ⋅ P(B)
Conditional Probability	P(A | B) = P(A∩B) / P(B)
Bayes Formula	P(A | B) = P(B | A) ⋅ P(A) / P(B)

Probability Tree Diagram

Types of Probability

Classical Probability

Classical probability is considered to be a simple type of probability that is based on the equal odds of the occurrence of an event. For example : Rolling a fair die. It's equally likely that we can get a 1, 2, 3, 4, 5, or 6.

Empirical Probability

Empirical probability is also termed as the experimental probability that refers to the probability which is mainly based on the historical data. This type of probability depicts likelihood of the event occurrence based on the historical data.

Subjective Probability

Subjective probability is considered to be the type of probability that is derived from the personal judgement of an individual or his or her own experience.

Axiomatic Probability

Axiomatic probability is considered to be a unifying theory of probability. The probability sets down the set of rules or axioms which can be applied to different types of probability.

Finding the Probability of an Event

The probability of a particular event depicts likeliness of an event to occur. The probabilities of various events are written as decimals or fractions. For example, we can consider a fruit bowl that contains five pieces of fruit including three bananas and two apples.
If a person wants to select one piece of fruit to eat for a snack and he or she does not care what it is, there is a 3535 probability that the person will choose a banana as there are three bananas out of the total of five pieces of fruit. The probability of an event is the number of favourable outcomes divided by the total number of outcomes.

Probability of choosing a banana = 3/5

Coin Toss Probability

Probability is considered to be a measurement of chances based on the likelihood of the occurrence of an event. If probability of occurrence of the event is high, it is likely that an event will happen. The probability of a coin toss can be calculated in the following manner,

Number of possible outcomes = 2

Number of outcomes to get head = 1

Probability of getting a head = ½

Hence, Probability of getting ahead = No of outcomes to get head/No of possible outcomes

Dice Roll Probability

Dice roll probability is quite common in statistics and probability and the individuals gain wide variety of results for the six-sided dice. For example, five and a seven, a double twelve, or a double-six. 

Probability of Drawing Cards

A card deck consists of four Aces, thirty-six number cards, twelve face cards within a fifty-two-card based deck. The probability of drawing any card will always lie between zero and one. The number of spades, hearts, diamonds, and clubs is same in every pack of 52 cards.

Probability Theorems

Bayes' Theorem on Conditional Probability

Bayes’ theorem is named after a British mathematician of the 18th century named Thomas Bayes for the purpose of determining the conditional probability. The conditional probability is likelihood of the outcome that can occur based on the occurrence of previous outcome. Bayes' theorem is considered to be a formula which aims at describing the ways of updating the probabilities of the hypotheses when evidence is given. This mainly follows from the axioms of conditional probability; however, it can be used for the purpose powerfully reasoning about the vast range of problems that also involve the belief updates.

For example, given a hypothesis HH and evidence EE, Bayes' theorem states that the relationship between the probability of the hypothesis before getting the evidence P(H)P(H) and the probability of the hypothesis after getting the evidence P(H mid E)P(H∣E) is,

P(H E) = frac{P(E mid H)} {P(E)} P(H).P(H∣E)=P(E)P(E∣H)​P(H).

The techniques of modern machine learning can rely on the Bayes’ theorem and can be used for calculation of probability.